Definition of an appropriate free dynamics and the physical S-matrix in multichannel hyperradial adiabatic scattering

¹ Bogoliubov Laboratory of Theoretical Physics , Joint Institute for Nuclear Research, 141980 Dubna, Moscow, Russia
² Kitasato University, 1-15-1 Kitasato, Sagamihara, Kanagawa 228-8555, Japan
³ Institut für Physik, Universität Mainz, D-55099, Mainz, Germany

^a e-mail: amatv@theor.jinr.ru

Abstract

In the hyperradial adiabatic (HA) treatment of the three-body problem the total wave function is i expanded as Ψ_HA(R, ξ, η) = R^−5/2 ∑_iχ_i(R)φi(R|ξ, η),where R denotes the hyperradius and (ξ , η) are internal hyperangles. Integration over ξ and η converts the Schrödinger equation into a system of coupled hyperradial equations. It is a well-known fact that, within the HA approach, the non-adiabatic corrections that couple channels converging to the same asymptotic configuration can show an unphysical long-range behavior ∼ 1/R. Though the latter is of purely kinematic origin and arises from the use of the hyperradius instead of the pertinent Jacobi variables, it is nevertheless the source of the considerable difficulties inherent in this approach. Here we propose, following the analysis of [1, 2], to define appropriate hyperradial-distorted free incoming and outgoing waves (H DF W ) that incorporate these unphysical long-range effects. Using them the physical S−matrix can be found in a straightforward manner.